Solve the following quadratic equation:

\[x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2}i = 0\]

#### Solution

\[ x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2} i = 0\]

\[\text { Comparing the given equation with the general form } a x^2 + bx + c = 0, \text { we get }\]

\[a = 1, b = - \left( \sqrt{2} + i \right) \text { and } c = \sqrt{2}i\]

\[x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]

\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} + i \right)^2 - 4\sqrt{2}i}}{2}\]

\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{1 - 2\sqrt{2} i}}{2} \]

\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} \right)^2 - 1^2 - 2\sqrt{2} i}}{2}\]

\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} - i \right)^2}}{2}\]

\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \left( \sqrt{2} - i \right)}{2}\]

\[ \Rightarrow x = \sqrt{2}, i \]

\[\text { So, the roots of the given quadratic equation are } \sqrt{2} \text { and } i .\]